Chiming in with my experience on pull ups and elbow pain. The pull up has long been my pet lift. My greatest feat sometime around 15 years ago was 20 pull ups with an extra 50 lbs at around 165 lbs BW.
The main causes behind elbow pain during pull ups is not keeping a NEUTRAL WRIST and general weakness of the scapula, exposed through poor technique once fatigue sets in.
First the grip: once you cock your wrist forward (usually happens before initiating the pull) the load distribution changes for the worst: much more load is now placed on the joints of the wrist and elbow instead of the shoulder! Wonder why finger extensions are a great rehab/prehab movement for tennis elbow? It's because the wrist flexors relax as a result if reciprocal inhibition!
The fix is to get your technique down. First, USE THE THUMBS AROUND GRIP. Do NOT use the false thumbless grip! Two reasons for this: 1) the thumb allows you to grip the bar much harder, fortifying the wrist via irradiation; 2) the standard grip allows you to fight the tendency to flex the wrist when the grip fatigues. The false grip grants none of these. I also suggest gripping the bar further down the fingers closer to the palm as opposed to the fingertips, shortening the lever arm around the wrist.
Next, the shoulder. I'm of the opinion that prescribing high volume or high load pull ups to the average weekend warrior or keyboard commando is not the best idea. An analogy would be back squats for office workers or similar whom have hips with the suppleness of plywood. The issue is that the average joe commonly lacks the strength of the lower traps to keep the scapula glued to the ribcage throughout the lift. This weakness manifests as swaying during the ascent which, coupled with a weak wrist position, spells eventual disaster for the elbow (and for some, the shoulder).
Shoulder fix: by whatever means, strengthen the lower traps. I find a simple dowel lift lying on a slight incline, focusing on shoulders back and down, does the trick. But there are a million and one movements out there. Secondly, integrate the lower traps without actually doing pull ups with hanging leg raises. The shoulders should feel tightly packed down. Keep this position and do not sway during the entire set. The torso angle during the HLR is the same as during the pull up - as close to ramrod straight as possible.
These fixes should set you straight for life. Elbow pain, like any pain, is an indication of some kind of weakness, either muscular or technique driven. Fix the wrist and shoulder weakness, tighten up technique, and you will find that not only can you avoid pain, but the pull up will grant you far greater gains in grip and back strength!